sq. FD + sq. DB = sq. BF Almagest Ptolemy

Hello, everyone I am reading through the Almagest by Ptolemy and I have a question I hope someone can resolve for me. In Book I Chapter 10 of the Almagest, Ptolemy sets up a semicircle ABC with two right triangles on the diameter AC. To view the text just click the link below:

In this text Ptolemy proves sq. FD + sq. BD = sq. BF but when I try to do the math my calculations are off by 7/60^2 + 2/60^3 + 16/60^4 (or about 0.2%). If anyone can help me to improve the precision of my calculations it would be very much appreciated!

Thanks in advance for any answers!

Below you can find my calculations of the squares on FD, DB and BF using the sexagesimal (base-60) system employed by Ptolemy:

We shall attempt to prove sq. FD + sq. DB = sq. BF with the values given in Ch 10 Book I of the Almagest. The line FD subtends an arc of 36 degrees and is equal to 37 parts 4 minutes and 55 seconds of the 120 parts of the diameter of the circle ABC. (Found by using the Table of Chords in Ch 11 Book I of the Almagest).

Re: sq. FD + sq. DB = sq. BF Almagest Ptolemy

It seems pretty clear that |FD|^2 + |BD|^2 = |BF|^2, from Pythagoras. My guess is that your error is due to rounding, since you only go as far as seconds in converting degrees to "parts of the diameter of the circle." If line BD is 60 units, and hence DE is 30 units, then the line FD is about 37.08204 units, which converts using your technique to 37, 4', 55.3416''. By truncating it at 55" you have introduced a small error - about 0.00026%. By my calculations the square of FD is 1375.078, whereas the square of your value is 1375.071. Put into you notation, the more accurate result is 1375, 4, 39, 30, 20.88 versus yours of 1375, 4, 14, 10, 25. This same type of error affects BF as well.

However, even though your calculations have these rounding errors, the magnitude of the error is really not as bad as you state - the error of 0.00197 out of 70.5 is only about 0.0028%, so ignoring everything smaller than 1/60^2 is truly of minor consequence.

Now, I have a question for you - since I am not familiar with this technique, can you please explain in a bit more detail what you mean by: "The line FD subtends an arc of 36 degrees ... " Where is this angle of 36 degrees?